Some Topics On Fractional Hamiltonian Systems

Fractional calculus has a history of more than 300 years,but until the last 20 years,with the continuous development of information technology,fractional calculus has at-tracted more and more attention,especially its unique memory characteristics,making fractional calculus play an important role in the field of physical engineering.Although there are many researches on fractional order differential system,so far,there are few re-searches on fractional order Hamiltonian system,and even there is no uniform definition of fractional order Hamiltonian system.Therefore,it is of great theoretical and practical significance to establish the theoretical system of fractional Hamiltonian systemThis thesis is divided into four parts,which mainly study three aspects,including the derivation of the fractional Hamiltonian equation,the stability of the fractional linear Hamiltonian system,and the analytical solution of the fractional Hamiltonian system with bilateral derivativesIn the first chapter we briefly introduce the development of fractional calculus,the background of fractional Hamiltonian system and the definitions and properties of frac-tional calculus.The second chapter,as one of the main contents of this paper,firstly summarizes the different forms of fractional order Hamiltonian equation obtained by different methods,and analyzes the relationship between different forms of fractional order Hamiltonian equation;secondly,it corrects the mistake made by predecessors in directly fractional order generalization of Hamiltonian equation,and gives a ?-order integral formula for 1<?<2,and through which,the ?-order Hamiltonian equation for 1<?<2 is derived.Finally,a method of deriving the fractional order Hamiltonian equation with only one side derivative is given,and the ?-order Hamiltonian equation with only one side derivative and ??(0,2)is derivedOn the basis of the second chapter,the third chapter studies the stability of a kind of fractional order Hamiltonian system(only with one-sided derivative),and gives the conditions for the asymptotic stability and stability of the ?-order linear Hamiltonian system with 0<?<1,and the conclusion that the solution of the ?-order linear Hamiltonian system with 1<?<2 is unstable.After that,for different normal forms of fractional order linear Hamiltonian systems,the stability analysis is carried out by using the above stability criteria theorems,and the solution and stability analysis are carried out for fractional Order Linear Nonhomogeneous Hamiltonian system.In Chapter four,another form of fractional Hamiltonian system with two-sided derivatives is discussed.Firstly,the general solution of differential equation with Riemann-Liouville fractional right derivative is solved by the method of step-by-step approxima-tion.Secondly,the analytic solution of fractional Hamiltonian system with two-sided derivatives in different cases is solved by this method.